We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation models, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal–dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions on the functionals—but still with a non-bilinear coupling term—we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems.

我们证明了困难的非凸非光滑优化问题，例如纳什均衡问题和各向异性以及各向同性的Potts分割模型，可以用凸泛函的广义​​共轭来表示。这些反过来又可以表述为涉及凸非光滑函数和一般光滑但非双线性耦合项的鞍点问题。然后，我们通过详细的收敛分析表明，Chambolle和Pock的原始对偶近端分裂方法在概念上的直接扩展适用于此类问题的解决。在功能上有足够的局部强凸假设的情况下，但仍具有非双线性耦合项，我们甚至证明了该方法的局部线性收敛。